Clear[l,k]

u0[x_]=100

u0dotsin=Integrate[u0[x]*Sin[m Pi/l x],{x,0,l}]

u0dotsin=Simplify[u0dotsin,Assumptions->Element[m,Integers]]

sindotsin=Integrate[Sin[m Pi/l x]^2,{x,0,l}]

sindotsin=Simplify[sindotsin,Assumptions->Element[m,Integers]]

b[m_]=u0dotsin/sindotsin

l=1;
Plot[Sum[b[m]*Sin[m Pi/l x],{m,1,100}],{x,0,l}]

l=1;
Plot[Sum[b[m]*Sin[m Pi/l x],{m,1,100}],{x,0,l},PlotRange->{0,120}]

Clear[l,k];
u[x_,t_,max_]:=Sum[b[n]*Exp[-(n Pi/l)^2k t]*Sin[n Pi/l x],{n,1,max}]

l=1;
k=1;
Plot[u[x,0.001,100],{x,0,l},PlotRange->{0,120},AxesLabel->{x,u}]

Manipulate[
Plot[u[x,t,100],{x,0,l},PlotRange->{0,120},AxesLabel->{x,u}],
{t,0,0.4,0.01}]

Plot[{u[0.5,t,100],u[0.5,t,1]},{t,0,0.2},PlotRange->{0,130},AxesLabel->{t,u}]